In order to evaluate the SINR, we first calculate the power of the received signal, which is calculated conditioned on the knowledge of the estimates \(\mathbf {z} \triangleq \text{vec} (\hat{\mathbf {H}}_t^k)\)  as \(\mathbb {E}_{\mathbf {z}} [|\tilde{y}_{tm}^{k}|^2] = \mathbb {E}_{\mathbf {z}} [| \sum _{i=1}^4 T_i|^2]\) .
Since noise is uncorrelated with data, \(\mathbb {E}_{\mathbf {z}} [T_1 T_4^H] = \mathbb {E}_{\mathbf {z}} [T_2 T_4^H] = \mathbb {E}_{\mathbf {z}} [T_3 T_4^H] = {0} \) .
Since MMSE channel estimates are uncorrelated with their errors [1]}, \(\mathbb {E}_{\mathbf {z}} [T_1 T_2^H] = {0}\) .
Computing the remaining power components requires the evaluation of \(\mathbb {E}_{\mathbf {z}} [x_i x_j]\)  for \(i \ne j\)  which can be calculated as \(\mathbb {E}_{\mathbf {z}} [x_i x_j] = \mathbb {E}_{\mathbf {z}} [x_i] \mathbb {E}_{\mathbf {z}} [x_j] = 0\) .
Thus, all the four terms are uncorrelated and the power in the received signal is just a sum of the powers of the individual components \(\mathbb {E}_{\mathbf {z}} [|\tilde{y}_{tm}^{k}|^2] = \sum _{i=1}^4 \mathbb {E}_{\mathbf {z}} [|T_i|^2]\) .
We now compute the powers of each of the components. The useful signal power is
\(\mathbb {E}_{\mathbf {z}} [|T_1|^2] &= \mathbb {E}_{\mathbf {z}} [|{\mathbf {a}}_{tm}^{kH} \hat{\mathbf {h}}_{tm}^{k} g_{tm} x_{m}|^2] = P g_{tm}^2 |{\mathbf {a}}_{tm}^{kH} \hat{\mathbf {h}}_{tm}^{k}|^2.\) 
